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A new multi-target filtering algorithm, termed as the Gaussian sum probability hypothesis density (GSPHD) ... (NN), joint probabilistic data association (JPDA),.
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Chinese Journal of Aeronautics 21(2008) 341-351

Journal of Aeronautics

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Gaussian Sum PHD Filtering Algorithm for Nonlinear Non-Gaussian Models Yin Jianjun, Zhang Jianqiu*, Zhuang Zesen Department of Electronic Engineering, Fudan University, Shanghai 200433, China Received 6 September 2007; accepted 26 April 2008

Abstract A new multi-target filtering algorithm, termed as the Gaussian sum probability hypothesis density (GSPHD) filter, is proposed for nonlinear non-Gaussian tracking models. Provided that the initial prior intensity of the states is Gaussian or can be identified as a Gaussian sum, the analytical results of the algorithm show that the posterior intensity at any subsequent time step remains a Gaussian sum under the assumption that the state noise, the measurement noise, target spawn intensity, new target birth intensity, target survival probability, and detection probability are all Gaussian sums. The analysis also shows that the existing Gaussian mixture probability hypothesis density (GMPHD) filter, which is unsuitable for handling the non-Gaussian noise cases, is no more than a special case of the proposed algorithm, which fills the shortage of incapability of treating non-Gaussian noise. The multi-target tracking simulation results verify the effectiveness of the proposed GSPHD. Keywords: signal processing; Gaussian sum probability hypothesis density; simulation; nonlinear non-Gaussian; tracking

1 Introduction1 The main objective of multi-target tracking is to jointly estimate the unknown and time-varying number of targets as well as their individual states from the history of noisy and cluttered observation sets. Most approaches to this problem involve data association techniques such as nearest neighbor (NN), joint probabilistic data association (JPDA), and multiple hypothesis tracking (MHT)[1-5], which constitute the bulk of the computational work in multi-target tracking algorithms. Finite set statistics (FISST) provides a general systematic foundation for multi-target filtering based on the theory of random finite set (RFS), which performs filtering on set-valued observations

*Corresponding author. Tel.: +86-21-55664227. E-mail address: [email protected] Foundation item: National Natural Science Foundation of China (60572023)

and states without explicit connections among measurements and targets[6-16]. RFS considers sets as elements, which can be seen as the extension of the random variable and random vector. In simple single-target tracking, where there are no appearing or disappearing targets or spurious measurements (clutter), the states and measurements are both vectors, whose dimensions will not submit to changes. However, in multi-target tracking, the number of targets and the measured tracks will be time-varying with changing dimensions of the states and measurements, because of targets disappearing, spawning, spontaneous births, and clutter. By modeling the collection of individual targets as an RFS and the collection of individual observations as another RFS, the problem of dynamically estimating multiple targets in the presence of clutter and associated uncertainty can be cast in a Bayesian filtering framework [7,9,16]. Such a theoretically optimal approach

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Yin Jianjun et al. / Chinese Journal of Aeronautics 21(2008) 341-351

to multi-target tracking is an elegant generalization of the single-target Bayesian filter, and circumvents the data association procedure. Mahler may have been the first person to introduce the RFS approach of multi-target tracking to the tracking community more than a decade ago[6-7]. Despite being theoretically solid, the RFS approach has been rejected by several tracking researchers and engineers owing to the involved intractable computations[10]. Recently, the RFS formulation has drawn considerably more attention thanks to the enhancement of the computational capability and the developed computational approximation strategies: particle filter (PF), also known as sequential Monte Carlo (SMC) sampling algorithms[17-20], and the probability hypothesis density (PHD) approximation[12-13,16]. SMC implementations of the RFS multi-target filtering recursion can be found in Ref.[9]. However, this method is still computationally intensive owing to the combinatorial nature of the densities, especially when the number of targets is large[16-17]. The idea of Rao-Blackwellisation is applied in Ref.[10] to reduce the computational loads, where only some states are sampled, while the others are handled analytically. As a novel RFS based filter, the PHD filter is a suboptimal but computationally tractable alternative to the RFS Bayesian multi-target filter[12-13,16]. It is a recursion that propagates the first-order statistical moment, or intensity, of the states RFS in time. One such algorithm, known as the SMCPHD (or particle PHD) filter, was proposed in Ref.[9] and Ref.[13], which used the SMC technique to propagate the posterior intensity in time. The main drawbacks of this approach are the large number of particles and the unreliability of clustering techniques for extracting the state estimates[16]. Another PHD algorithm, known as the Gaussian mixture PHD (GMPHD) filter, provides an analytic solution to the PHD recursion for linear Gaussian target dynamic model by approximating the intensity function with a weighted mixture of Gaussians[16]. The GMPHD algorithm is also extended to nonlinear target dynamic models using approximation strategies from

the extended and unscented Kalman filters. Since these techniques are all based on the Gaussian process and measurement noises, these may not be adequate to handle non-Gaussian models, which are more universal in practice. This article proposes a solution to the PHD recursion for nonlinear non-Gaussian dynamic model, termed as the Gaussian sum probability hypothesis density (GSPHD) filter, a generalization of the GMPHD algorithm. It is perceived that under the conditions where the state noise, the measurement noise, target spawn intensity, new target birth intensity, target survival probability, and detection probability are expressed as Gaussian sums, the predictive and posterior intensities at any subsequent time step remain Gaussian sums if the initial prior intensities are Gaussian or a Gaussian sums. Simulation results are presented to demonstrate the validity of the proposed filter.

2 Model Description 2.1 Models for random vector filtering Consider the general system model[21] as follows xk zk

f k ( xk 1 )  wk hk ( xk )  vk

(1) (2)

where xk and zk denote the state and measurement, respectively, wk and vk denote the process noise and the measurement noise, respectively, and both f k and hk denote the known nonlinear functions. Let x1:k { x1 ,", xk } and z1:k {z1 ," , zk } . Here, the purpose is to estimate the posterior probability density function, p( xk | z1:k ) , using the following equations p( xk | z1:k 1 )

³ f k|k 1 ( xk | xk 1 ) pk 1 ( xk 1 | z1:k 1 )dxk 1

(3)

p( xk | z1:k )

p ( zk | xk ) p ( xk | z1:k 1 )

³ p( zk | xk ) p( xk | z1:k 1 )dxk

(4)

2.2 Models for RFS filtering Suppose that the number of targets and measurements are M (k ) and N (k ) , respectively, at the time step k . Then, RFSs X k and Z k can be

Yin Jianjun et al. / Chinese Journal of Aeronautics 21(2008) 341-351

used to denote the multi-target states and measurements, respectively, as follows (5) X k { xk ,1 ," , xk ,M ( k ) } Zk

{zk ,1 ," , zk , N ( k ) }

(6)

Let pS,k ( xk 1 ) denote the probability when a target still exists at time k and its previous state xk 1 , f k |k 1 ( xk | xk 1 ) the state transition density. Then, for a given state xk 1  X k 1 at time k  1 , its behavior at the next time step is modeled as the RFS, i.e., ­{ x }, Survival (7) Sk |k 1 ( xk 1 ) ® k ¯‡, Disappear In this way, the target states at time k can be described as the union of survival, spawned, and spontaneous birth targets by Xk

ª º ª º «] *X S k |k 1 (] ) » * «] *X Gk |k 1 (] ) » * Bk ¬ k 1 ¼ ¬ k 1 ¼

(8)

where Gk |k 1 (] ) denotes the RFS of targets spawned from a target with previous state ] , and Bk denotes the RFS of the spontaneous birth targets. Assume that pD,k ( xk ) denotes the target detection probability, and hk ( zk | xk ) the probability density obtained by an observation zk of the state xk . Therefore, at time k , each state xk  X k generates an RFS as ­{ xk }, Detected ¯‡, Missed

4 k ( xk ) ®

(9)

The measurement Z k is formed by the union of targets generated by measurements and clutter, i.e., ª º (10) Zk « * 4 k ( x)» * M k x X  ¬ ¼ k

where M k denotes the false measurements or clutter. Let pk (